I am following my university course notes on amplification in laser media, and have come across expressions for the gain of a medium, but the notes are not exactly rigorous... The expression given for the gain $G$ in units of dB m$^{-1}$ is:

$$G = 10 \log_{10}(e^\gamma) \mathrm{dB} \mathrm{m}^{-1}$$ where $\gamma$ is also the gain coefficient for the media with units m$^{-1}$. However later in the notes it gives the expression $$\gamma = \Delta N \sigma$$ where $\Delta N$ is the "number of atoms in population inversion" and $\sigma = [\mathrm{m}]^{-2}$ is the "cross-section for stimulated emission". This clearly shows that $\Delta N = [\mathrm{m}]$, which doesn't make much sense to me. Also the first expression implies that $\gamma$ is unitless, so I am mighty confused.

1 Answer
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Bonus points for checking your dimensions. Your forumlas omit a lot of constants and assumes no atoms in the ground state, but that does not really matter for getting the units right. For that you need to realize that:

The unit for cross section is meter squared [m]x[m], not the reciprocal.

When you take e to the power of something, the "something" has to be dimensionless. Your formula for the gain omits the length traversed by the light. It should be (coefficient x length) which is unitless.

Finally you have forgotten the volume in the number of atoms. It should be number of atoms per volume (otherwise a twice as big laser would have a twice as high gain coefficient).